Then, we investigated the vanishing viscosity limit of the statistical solutions of the incompressible Navier-Stokes equations. To this end, we derived a suitable statistical version of the well-known Kármán-Howarth-Monin relation and used it to prove precise rates for the asymptotic decay of a averaged third-order structure function in Lemma REF . However, these estimates do not suffice to pass to the \(\varepsilon \rightarrow 0\)  limit. To this end, we assumed a weak statistical scaling of the Navier-Stokes statistical solutions (see Assumption REF ). This assumption is a weaker version of Kolmogorov's scaling assumptions in his K41 theory. Moreover, it is consistent with and incorporates different variants of scaling that are proposed in the physics literature to explain intermittent corrections to Kolmogorov's theory. Under this assumption, we proved a weak anisotropy result and invoked compactness results of [1]} to rigorously prove that the statistical solutions of the Navier-Stokes equations converge, in a suitable sense, to a statistical solution of the incompressible Euler equations. Thus, we were able to characterize the vanishing viscosity limit of the Navier-Stokes equations in a relevant regime.
